Since you aren't putting much weight on it, I think I'd try wire guides of the sort that wrap around the wire and are then secured to the having material with a screw. How many you'd want would depend on the planned load. This would look a bit more official/permanent than zip ties. These are sometimes available in dark colors, if you can find those they'd make the anchoring points less obvious.
But this is the sort of project where it's often simplest to go into a hardware store, see what they have in the way of mounting hardware, and pick out something you think will work and not look bad. You're unlikely to go very far wrong; at worst you might miss a more elegant solution but almost anything that can grip both grid and uprights should work.
Heck, you could just drive an angled nail on each side and hang it from them, poor-man's-picture-hook style.